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We establish local-in-time and global-in-time well-posedness for small data, for a coupled system of nonlinear acoustics-structure interactions. The model consists of the nonlinear Westervelt equation on a bounded domain with nonhomogeneous boundary conditions, coupled with a fourth-order linear equation defined on a lower dimensional interface occupying a part of the boundary of the domain, with transmission boundary conditions matching acoustic velocities and acoustic pressures. While the well-posedness of the Westervelt model has been well studied in the literature, there has been no works in the literature on well-posedness of acoustics-structure interaction models involving the Westervelt equation. Another contribution of this work, is a novel variational weak formulation of the linearized system and a consideration of various boundary conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well-posedness and long-time dynamics of a class of beam/plate equations with rotational inertia and nonlinear energy damping. The model is derived from nonlocal dissipative energy models for flight structures, as proposed by Balakrishnan-Taylor (Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). Our main results address the existence of compact global attractors. The work complements the degenerate coefficient case left open by Sun and Yang (J. Math. Anal. Appl., Volume 512, Issue 2, 2022). [ABSTRACT FROM AUTHOR]

This paper deals with the infinite blow‐up phenomena for a class of damped plate equations with logarithmic nonlinearity under the Navier boundary condition. Combining potential well method and modified differential inequality technique, we establish the infinite blow‐up result of solutions with arbitrary initial energy. In particular, it is not necessary to suppose that the initial velocity and the initial displacement should have the same sign in the sense of the L2${L^2}$ inner product, that is, the solution may blow up at infinity even ∫Ωu0u1dx<0$\int _\Omega {{u_0}{u_1}dx} < 0$, more precisely, ∫Ωu0u1dx>−12u0∗2$\int _\Omega {{u_0}{u_1}dx} > - \frac{1}{2}\left\Vert {{u_0}} \right\Vert _*^2$. [ABSTRACT FROM AUTHOR]

The dynamics of solutions for the plate equation without delay effects has been investigated by many authors. But there are few works on plate equation with variable delay on unbounded domain. Therefore, we firstly consider the existence of pullback attractor for the plate equation with variable delay on by using the theory of multivalued dynamical system. [ABSTRACT FROM AUTHOR]

Based on the existence of pullback attractors for the non-autonomous stochastic plate equations with multiplicative noise and nonlinear damping defined in the entire space Rn by Xiaobin Yao in [15], in the paper, we further investigate the upper semicontinuity of pullback attractors for the problem. [ABSTRACT FROM AUTHOR]

In the paper, the numerical solution to the linearized problem of a partially clamped, partially free rectangular plate is considered in order to model and simulate the unstable end behavior. Such a problem is modeled by a nonlinear plate evolution equation with stretching effects in the span-wise direction and wind-flow in the chord-wise direction. The motivation is to model a cantilever bridge in order to consider the long-time behavior of a nonlinear elastic structure forced by external/internal inputs. The non-conservative force acting upon the system and the resulting loss of dissipativity of the system are the major features. The dependence of the structure of stationary states on the values of the gas velocity will be analyzed. A numerical approximation with a Morley finite element is adapted to the plate with clamped-free edges, and corresponding numerical computations are provided. [ABSTRACT FROM AUTHOR]

In this paper, we examine the problem on the regional optimal control of a vibrating plate in a spatial domain Ω. We obtain a bounded control that drives such a system from an initial state to a desired state in a finite time, only on a subdomain ω of Ω. We prove that a regional optimal control exists characterize this control. Also we propose a condition that ensures the uniqueness of an optimal control and develop an algorithm for numerical simulations. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the following weakly damped nonlinear suspensionbridge equation utt(x, y, t) + ∆2u(x, y, t) + α(t)g(ut) = 0, and establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions α and g and obtained without any restriction growth assumption on g at the origin. The multiplier method, the properties of the convex and the dual of the convex functions, Jensen’s inequality and the generalized Young inequality are used to establish the stability results. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates mainly the asymptotic behavior of the nonautonomous random dynamical systems generated by the plate equations driven by colored noise defined on R n $\mathbb{R}^{n}$ . First, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.

We consider the dynamic elasticity equation, modeled by the Euler Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic) damping in a boundary domain. Using a frequency domain method based on the Batty and Duyckaerts result [7], combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity. [ABSTRACT FROM AUTHOR]

In a bounded domain, we consider a viscoelastic Euler–Bernoulli equation called also viscoelastic plate equation, with logarithmic nonlinearity source in the right-hand side, u tt + Δ 2 u - ∫ 0 t g (t - s) Δ 2 u (s) d s + h (u t) = | u | γ - 2 u ln | u | , where γ > 2 and the relaxation function g satisfied g ′ (t) ≤ - ξ (t) H (g (t)) , ∀ t > 0 , where H is an increasing and convex function near the origin and ξ is a nonincreasing function. In present of a nonlinear feedback on a part of the boundary, we establish, with certain initial data, a general decay results, using the multiplier method and some properties of the convex functions. Our new results generalize and significantly improve earlier results in the literature, in particular, the result of Al-Gharabli et al. (Commun Pure Appl Anal 18(1): 159–180, 2019), Al-Gharabli et al. (Math Comput Appl 27: 10, 2022), Mustafa (Evol Equ Control Theory 6(2): 61–276, 2017) and Cavalcanti et al. (J Differ Integral Equ 17: 495–510, 2004). [ABSTRACT FROM AUTHOR]

This paper focuses on an inverse problem associated with the plate equation which is derived from models in fluid mechanics and elasticity. We establish the unique identifying results in simultaneously determining both the unknown density and the internal sources from the passive boundary measurement. The proof mainly relies on the asymptotic analysis and harmonic analysis on integral transforms. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the following nonlinear plate equation $ u_{tt}+\Delta ^2 u +\alpha(t) g(u_t) = u \vert u\vert ^{\beta}, $ which models suspension bridges. Firstly, we prove the local existence using Faedo-Galerkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions $ \alpha $ and $ g $ and obtained without any restriction growth assumption on $ g $ at the origin. The multiplier method, properties of the convex functions, Jensen's inequality and the generalized Young inequality are used to establish the stability results.

The paper investigates mainly the asymptotic behavior of the non-autonomous random dynamical systems generated by the plate equations with memory driven by colored noise defined on $ \mathbb{R}^n $. Firstly, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.

This paper investigates mainly the asymptotic behavior of the nonautonomous random dynamical systems generated by the plate equations driven by colored noise defined on R n . First, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the asymptotic behavior of a coupled system that describes the vertical displacement of thin Kirchhoff plates. One of the equations has a viscoelastic dissipation of memory type and the other is conservative. The memory depends on a fractional operator with exponent θ ∈ [ 0 , 1 ] and its kernel is an exponentially decreasing function. We obtain explicit optimal polynomial decay rates for the solutions with regular initial data. The decay rates depend on the exponent of the memory and some relations between the mass density, the flexural rigidity coefficients and the coefficients of the rotational inertia terms. Considering equal mass densities, the worst decay rate happens when the rotational inertia coefficients are distinct, while the best decay is only possible when the rotational inertia coefficients coincide as well as the flexural rigidity coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the general decay rate of the solutions for a class of plate equations with memory term in the whole space Rn, n ≥ 1, given by utt + Δ²u + u + ∫t0 g(t - s)Au(s)ds = 0; with A = Δ or A = -Id. We use the energy method in the Fourier space to establish several general decay results which improve many recent results in the literature. We also present two illustrative examples by the end. [ABSTRACT FROM AUTHOR]

The main goal of this work is to investigate the following nonlinear plate equation u t t + Δ 2 u + α (t) g (u t) = u | u | β , which models suspension bridges. Firstly, we prove the local existence using Faedo-Galerkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions α and g and obtained without any restriction growth assumption on g at the origin. The multiplier method, properties of the convex functions, Jensen's inequality and the generalized Young inequality are used to establish the stability results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of stochastic plate equations with memory in unbounded domains. More specifically, we obtain the uniform time estimates for solutions of the problem. Based on the estimates above, we prove the existence and uniqueness of random attractors in unbounded domains.

In this paper, we consider Carleman estimates for the damped fourth-order plate operators ∂ t t − ρ ∂ t Δ + Δ 2 in a bounded smooth domain with Robin boundary conditions. Because the appearance of such kind of structural damping, the speed of propagation for solutions to the plate equation is infinite and the corresponding properties of the solution similar to heat equations, and is significantly different from that of the usual plate equations without damping. As applications, we consider two types of inverse problems in determining source terms for the plate equation with structural damping. The time-dependent measurements can be restricted to an arbitrary small sub-domain or arbitrary sub-boundary, respectively. Under some assumptions on the regularity of the solutions and coefficients, we prove the global stability results for these inverse problems. [ABSTRACT FROM AUTHOR]

This article considers a coupled system of non-linear plate equations with partially hinged boundary conditions. By assuming general conditions on the relaxation functions, we show that the associated energy functional satisfy a general decay estimate. The importance of this work is the extension and generalization of a considerable number of results in the case of wave equations to that of plate equations.

In this paper, we study the interplay between a fractional damping (− Δ)θut with θ ∈ [0, 1/2), and a nonlinear memory term applied to a plate equation: utt−Δutt−Δu+Δ2u+(−Δ)θut=∫0t(t−s)−γ|u(s,·)|pds,t>0,x∈ℝn,in space dimension n=1,2,3,4. In different scenarios, we prove the global existence of small data solutions in C([0,∞),H2)∩C1([0,∞),H1) for supercritical powers, where the critical exponent is determined by the fractional power of the damping term and by the order of the fractional integration in the memory term. In one case, we also feel the influence of the regularity‐loss type decay, which is due to the presence of the rotational inertia term −Δutt in the plate equation. [ABSTRACT FROM AUTHOR]

The initial-boundary value problem for a plate equation with a nonlocal source and singular nonlocal damping terms is considered. By using the multiplier method and weighted integral inequalities, we prove that the energy decays exponentially when the damping term has a certain singular nonlinearity. The results of this paper improve the earlier results. [ABSTRACT FROM AUTHOR]

In this paper, the blow-up of solutions for the following Dirichlet-Neumann problem to initial nonlinear viscoelastic plate equation with a lower order perturbation of p⃗ (x,t)p→(x,t)-Laplacian operator in the presence of time delay is obtained utt+Δ2u+Δp⃗ (x,t)u−∫t0g(t−s)Δ2u(s)ds−μ1Δut−μ2Δut(t−τ)=u|u|q−2.utt+Δ2u+Δp→(x,t)u−∫0tg(t−s)Δ2u(s)ds−μ1Δut−μ2Δut(t−τ)=u|u|q−2. Under suitable conditions on gg and the variable exponent of the p⃗ (x,t)−p→(x,t)− Laplacian operator, we prove that any weak solution with nonpositive initial energy as well as positive initial energy blows up in a finite time.

This paper investigates a regional optimal control problem of a plate equation described by a bilinear system evolving in a spacial domain Ω. The control is distributed, bounded and acts on the velocity term. Then, we minimise a functional cost constituted of the deviation between a desired state and the reached one only on a subregion ω of Ω and the energy term. The purpose of this study is to prove that a control solution of such problem exists, and characterised as a solution to an optimality system. Numerical approach is given and successfully illustrated by simulations. [ABSTRACT FROM AUTHOR]

This paper is devoted to the lifespan of solutions to a damped plate equation with logarithmic nonlinearity $ u_{tt}+\Delta^2u-\Delta u-\Delta u_t+u_t = |u|^{p-2}u\ln|u|. $ Finite time blow-up criteria for solutions at both lower and high initial energy levels are established and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned with the decay rate of the solution of a viscoelastic plate equation with infinite memory and logarithmic nonlinearity. We establish an explicit and general decay rate results with imposing a minimal condition on the relaxation function. In fact, we assume that the relaxation function h satisfies h ′ ( t ) ≤ − ξ ( t ) H ( h ( t ) ) , t ≥ 0 , $$ h^{\prime}(t)\le-\xi(t) H\bigl(h(t)\bigr),\quad t\geq0, $$ where the functions ξ and H satisfy some conditions. Our proof is based on the multiplier method, convex properties, logarithmic inequalities, and some properties of integro-differential equations. Moreover, we drop the boundedness assumption on the history data, usually made in the literature. In fact, our results generalize, extend, and improve earlier results in the literature.

Based on the abstract theory of pullback attractors of non-autonomous non-compact dynamical systems by differential equations with both dependent-time deterministic and stochastic forcing terms, which introduced by B. Wang, we investigate existence of pullback attractors for the non-autonomous stochastic plate equations with multiplicative noise defined in the entire space $\mathbb{R}^n$.

In this paper, we study the long-time behavior of the following plate equation ε (t) u t t + g (u t) + Δ 2 u + λ u + f (u) = h , where the coefficient ε depends explicitly on time, the nonlinear damping and the nonlinearity both have critical growths. [ABSTRACT FROM AUTHOR]

In this article, we study a plate equation with frictional damping, nonlinear source and time delay. Firstly, we establish the local existence by using the semigroup theory. Then, under suitable conditions, we prove the nonexistence of global solutions for positive initial energy. Time delays often appear in many practical problems such as thermal, economic phenomena, biological, chemical, physical, electrical engineering systems, mechanical applications and medicine. [ABSTRACT FROM AUTHOR]

We are concerned with asymptotic stability of energy for plate equations with vanishing viscoelastic dampings and complementary frictional dampings, where the internal viscoelastic dampings are assumed to be time-dependent. In terms of the relaxation function θ (t) and the coefficient γ (t) of viscoelastic term, we obtain the decay rates for the plate equations' energies. Since we do not need the viscoelastic damping coefficient to be bigger than a fixed positive number, which is required in previous investigations of the asymptotic stability of the plate equations' energies, our decay theorems extend and improve essentially the existing decay results for the plate equations in both viscoelastic damping case and mixed-type damping case. [ABSTRACT FROM AUTHOR]

In this paper, we consider a plate equation as a model for a suspension bridge with a general nonlinear internal feedback and time-varying weight. Under some conditions on the feedback and the coefficient functions, we establish a general decay estimate for the associated energy functional, from which the exponential and the polynomial, are particular cases.

We are concerned with dynamics of the weakly damped plate equation on a time-dependent domain. Under the assumption that the domain is time-like and expanding, we obtain the existence of time-dependent attractors, where the nonlinear term has a critical growth. [ABSTRACT FROM AUTHOR]

We consider a damped plate model with rotational forces in a bounded domain. The plate is either clamped or hinged. The rotational forces and damping involve the spectral fractional Laplacian with powers θ in [0, 1] and δ in [0, 2], respectively. Using the frequency domain approach, and appropriate interpolation inequalities, we show that the underlying semigroup is: (a) analytic for all θ in [0, 1] and δ in [ (2 + θ) / 2 , 2 ] , (b) not analytic for all θ in [0, 1] and δ in (θ , (2 + θ) / 2) , (c) of Gevrey class α > (2 - θ) / 2 (δ - θ) for all θ in [0, 1] and θ < δ < (2 + θ) / 2 . We also prove that for each admissible value of θ , the semigroup is exponentially stable for δ ≥ θ , and only polynomially stable, with rate O (t - 2 - θ 2 (θ - δ) ) , for δ < θ , when θ > 0 . In particular, in the hinged plate case, we show that the polynomial decay rate is optimal. [ABSTRACT FROM AUTHOR]

We consider a coupled viscoelastic plate nonlinear equations with degenerate damping terms on a bounded domain in Rn. We establish the global existence results under different conditions. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic profile (as t → ∞) of the solution to the Cauchy problem for the linear plate equation u t t + Δ 2 u − λ (t) Δ u + u t = 0 when λ = λ (t) is a decreasing function, assuming initial data in the energy space and verifying a moment condition. For sufficiently small data, we find the critical exponent for global solutions to the corresponding problem with power nonlinearity u t t + Δ 2 u − λ (t) Δ u + u t = | u | p. In order to do that, we assume small data in the energy space and, possibly, in L 1 . In this latter case, we also determinate the asymptotic profile of the solution to the semilinear problem for supercritical power nonlinearities. [ABSTRACT FROM AUTHOR]

In this article, we consider a nonlinear damped Petrovsky equation in bounded domain. The nonlinear damping is effective only in a neighborhood of a suitable subset of the boundary. The well‐posedness and regularity of solution is discussed owing to the nonlinear semigroup theory together with the Faedo–Galerkin approach. By energy method combined with the piecewise multiplied method and relying on the localized smoothing property, we show the exponential and polynomial stabilities by discussing with respect to the parameter p. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of Kirchhoff plates with intermediate damping. The damping considered contemplates the frictional and the Kelvin-Voigt type dampings. We show that the semigroup those equations decays polynomially in time at least with the rate t^{-1/(2-2θ)}, where θ is a parameter in the interval [0,1[. Moreover, we prove that this decay rate is optimal.

Abstract In this paper we study the viscoelastic plate equation with p-Laplacian and time-varying delay. We establish a general decay rate result under some restrictions on the coefficients of strong damping and strong time-varying delay and weakening the usual assumptions on the relaxation function, using the energy perturbation method.

In this work we study decay rates for a σ-evolution equation in R n under effects of a damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient, b (t) (− Δ) θ u t (t , x). We consider that b is 'confined' in the curve g (t) = (1 + t) α ln γ (1 + t) for large t ⩾ t 0 and without any control on d d t b (t). [ABSTRACT FROM AUTHOR]

We prove the existence and uniqueness of global solutions by Faedo-Galerkin method for the Cauchy problem concerning the evolution equation suggested by the study of plates and beams, where p ≥ 2 and Φ is a real function. We also investigate the asymptotic behavior of the solutions, under suitable growth assumptions. We will use for this task an appropriate perturbed energy coupled with multiplier technique. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for plate equations with rotational inertia and frictional damping terms. We derive asymptotic profiles of the solution in L 2 -sense as t → ∞ in the case when the initial data have high and low regularity, respectively. Especially, in the low regularity case of the initial data one encounters the regularity-loss structure of the solutions, and the analysis is more delicate. We employ the so-called Fourier splitting method combined with the explicit formula of the solution (high-frequency estimates) and the method due to [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differential Equations257 (2014) 2159–2177.] (low-frequency estimates). In this paper, we will introduce a new threshold l ∗ : = n / 2 − 1 on the regularity of the initial data that divides the property of the corresponding solution to our problem into two parts: one is wave-like, and the other is parabolic-like. [ABSTRACT FROM AUTHOR]

This article considers a dynamical system with delay described by a differential equation with partial derivatives of hyperbolic type and delay with respect to a time variable. We establish in Theorem 3.1 the k(t)-stability of weak solution under suitable initial conditions in by introducing appropriate Lyapunov functions. [ABSTRACT FROM AUTHOR]